Abstract
This document contains a fit to AiGauss(z) = erfc(z)/2 with an error  < 10-15 
AiGauss(z). #BestFit
The code is in ..\..\zips\FittingBrack.zip. #ZipFile is an annotated directory of this file.
AiGauss
The function is defined by
z
1
2
AiGauss  z  
 exp  t  dt (1.1)


This is approximated as
z  ...  z n
f A ( z )  exp   z 2 
 exp   z 2  Brack  z  (1.2)
1  ...  z n 1
Where
z  ...  z n
(1.3)
Brack  z  
1  ...  z n 1
The form in (1.2) has 15 digit accuracy for the exponential and also for the Pade
approximate. This is better than the form Pade = log( AiGau(z)) which has 15 digit
accuracy in the log, but uses two of these digits at z = -6 where the lot is approximately 36 to simply represent the order of magnitude of the function.
The first fit showed errors in the integral version of AiGauss itself
../aigau15/Fitting%20Brack.doc .htm
The code for producing AiGauss data is saved here as ..\..\zips\genint.zip Compiling was
done with
Wfl386 tgleg67
This produced a welcome extra digit of accuracy beyond that available with Watfor.
The end
Aigau2.dir
AIGAU.DIR, OUTPUT FILE NAME
..\TestData\BRACK.OUT,
DATA SET NAME
1,
DIMENSION OF X IN F(X)
8,8 NUMBER OF NUMERATOR TERMS, DENOMINATOR TERMS
0, 0 fits to exp(-x**2) times a Pade, 0 to a Pade
1E-25,1E-15, err(i)= max(ep1,ep2*dat(i))  this fit is to a part in
100000,
NUMBER OF MINIMIZATION STEPS
1015 – real*8 limit.
.9, INITIAL FR DECREASE DESIRED  as we lowe precision small jumps are
harder than big ones
1 VARY O FIX
FITTED CONS ERROR IN CONS
0 0.5000000000000000
+- 0.000E+00
1 -0.7780697362924373
+- 0.916E-04
1 0.5990638419338665
+- 0.129E-03
1 -0.2771270371606460
1 0.8027282341142808E-01
1 -0.1379770398066123E-01
1 0.1113061469828807E-02
1 0
1 -2.684518639716602
1
3.227282589432518
1 -2.263586666841055
1
1.006883612355409
1 -0.2865296587487073
1 0.4891172945745505E-01
1 -0.3945696943660320E-02
1 0
CHIS=0.3550E+08 FOR 5001 DATA
CHI USED IN CALCULATING ERRORS
++++-
0.891E-04
0.354E-04
0.805E-05
0.857E-06
+++++++-
0.183E-03
0.465E-03
0.520E-03
0.330E-03
0.127E-03
0.285E-04
0.304E-05
POINTS
IS 0.3550E+08
Figure 1 fi-fA(xi)
In this case fi is exp(z2)  AiGauss(z), and fA(x) = Brack(x) with the 8 numerator
and denominator constants shown above. The quantitiy minimized emphasizes the left
side of figure 1. The numerical thickness in the data to the left indicates how far the fit
has to go to get to the noise level.
I:\PUBLIC~1\nlfit\EXPTIM~1>mpnlfit aigau2.dir  the complete output is below, but this is
NADIR=aigau2.dir
a good time for coffee.
NAOUT=AIGAU.DIR
NADAT=..\TestData\BRACK.OUT
ND=
1
VORF
CONS
0 0.500000
1 -0.778070
1 0.599064
1 -0.277127
1 0.802728E-01
1 -0.137977E-01
1 0.111306E-02
1 0.000000E+00
1 -2.68452
1
3.22728
1 -2.26359
1
1.00688
1 -0.286530
1 0.489117E-01
1 -0.394570E-02
1 0.000000E+00
FIT IS BEING MADE TO DATA IN FILE
..\TestData\BRACK.OUT
FR,CHI,CHB,CHL 0.900000000
35499382.3
0.100000000E+67 0.500000000E+67
FR,CHI,CHB,CHL 0.729000000
25968696.2
25879049.7
35499382.3
FR,CHI,CHB,CHL 0.387420489
10184521.6
10060805.0
25968696.2
FR,CHI,CHB,CHL 0.581497370E-01 556637.656
592227.255
10184521.6
from bnraph going low  cannot get to .01 * chi with any lambda  exp(-175)
FR,CHI,CHB,CHL 0.100000000E-01 16867.8334
6673.35984
556637.656
from bnraph going low
FR,CHI,CHB,CHL 0.100000000E-01 9421.78133
9466.63002
16867.8334
from bnraph going low
FR,CHI,CHB,CHL 0.100000000E-01 9411.07375
9407.58683
9421.78133
from bnraph going low
FR,CHI,CHB,CHL 0.109000000
9423.80202
9409.60583
9411.07375
IREV=
1 IEND=99993
from bnraph going low
FR,CHI,CHB,CHL 0.910900000
9423.80202
9409.60583
9411.07375
IREV=
2 IEND=99993
from bnraph going low
FR,CHI,CHB,CHL 0.991090000
9423.80202
9409.60583
9411.07375
IREV=
20 IEND=99993  I modified the code, so that if it keeps getting the same value
it jumps to the last IREV – within 20-100, the chi-square
at this level is dominated by truncation errors.
AT END FR, CHI 0.999109000
9411.07375
TEMP2= 0.000000000E+00
NEW SECTION FR=
0.9991090000000000  nlfit is set to automatically restart
FR,CHI,CHB,CHL 0.999999990
9411.07375
0.100000000E+67 0.500000000E+67
FR,CHI,CHB,CHL 0.999999970
9411.07375
9411.07347
9411.07375
IREV=
1 IEND=99992
FR,CHI,CHB,CHL 0.999999997
9411.07375
9411.07372
9411.07375
IREV=
2 IEND=99992
FR,CHI,CHB,CHL 0.999999999
9411.07375
9411.07374
9411.07375
AT END FR, CHI 0.999999999
9411.07375
TEMP2= 0.999999901E-09
NEW SECTION FR=
0.9999999990000000
CHI AT END OF FIT
9411.0737524199120000
0
0.5000000000000000 +0.0000000000000000
1
-0.8591435477130216 +- 1.6272308649864600D-004
1
0.7341466366171062 +- 2.5735696428003250D-004
1
-0.3871217792735996 +- 2.0090444822930290D-004
1
0.1342351250044111 +- 9.4349896125745370D-005
1
-0.0304744298576865 +- 2.7806336792956480D-005
1
0.0042054633161981 +- 4.8818201328104450D-006
1 -2.7497424824452200D-004 +- 4.0523682417858010D-007
1
-2.8466662625211940 +- 3.2544617298786200D-004
1
3.6804121795519390 +0.0008819329591858
1
-2.8327305035912660 +0.0010715148057411
1
1.4258647627189870 +0.0007606586755338
1
-0.4833027070743471 +- 3.4312244331430270D-004
1
0.1085165756053468 +- 9.9288237674976140D-005
1
-0.0149079729638434 +- 1.7305634415493940D-005
1
0.0009747584530339 +- 1.4365264327455150D-006
FINAL CHIT=
9411.0737524199120000
CHI/DEG OF FREE
1.8874997497833760
RESULTS IN FILE AIGAU.DIR
The entire run takes about ½ hour on a 1 Giga-Hertz laptop.
Aigau.dir
aigau2.dir, OUTPUT FILE NAME
..\TestData\BRACK.OUT,
DATA SET NAME
1,
DIMENSION OF X IN F(X)
8,8 NUMBER OF NUMERATOR TERMS, DENOMINATOR TERMS
0, 0 fits to exp(-x**2) times a Pade, 0 to a Pade
1E-25,1E-15, err(i)= max(ep1,ep2*dat(i))
100000,
NUMBER OF MINIMIZATION STEPS
.9, INITIAL FR DECREASE DESIRED
1 VARY O FIX
FITTED CONS ERROR IN CONS
0 0.5000000000000000
+- 0.000E+00
1 -0.8591435477130216
+- 0.163E-03
1 0.7341466366171062
+- 0.257E-03
1 -0.3871217792735996
+- 0.201E-03
1 0.1342351250044111
+- 0.943E-04
1 -0.3047442985768653E-01
+- 0.278E-04
1 0.4205463316198079E-02
+- 0.488E-05
1 -0.2749742482445220E-03
+- 0.405E-06
1 -2.846666262521194
+- 0.325E-03
1
3.680412179551939
+- 0.882E-03
1 -2.832730503591266
+- 0.107E-02
1
1.425864762718987
+- 0.761E-03
1 -0.4833027070743471
+- 0.343E-03
1 0.1085165756053468
+- 0.993E-04
1 -0.1490797296384342E-01
+- 0.173E-04
1 0.9747584530339248E-03
+- 0.144E-05
CHIS= 9411.
FOR 5001 DATA POINTS
CHI USED IN CALCULATING ERRORS IS 9411.
Figure 2 (f-fA)/(1e-15*f)
The region -6<z<4 is approaching its accuracy limit. This is probably caused by
truncation errors in evaluating the integrals. To the extent that these effects are “random”
the underlying curve is more accurate by a factor of (Nd/Ncons) = 17. Both regions have
systematic errors on the order of 2 implying that this fit does not quite reach the goal.
The pass with 9,9 unlike the earlier ones, does not drop chi-square by orders of
magnitude. The numerical noise ~ 100 also stops a systematic approach to the minimum,
once its vicinity has been reached.
FIT IS BEING MADE TO DATA IN FILE
..\TestData\BRACK.OUT
FR,CHI,CHB,CHL 0.900000000
9612.87997
FR,CHI,CHB,CHL 0.729000000
7079.62515
from bnraph going low
FR,CHI,CHB,CHL 0.387420489
6713.67300
from bnraph going low
FR,CHI,CHB,CHL 0.152334763
6711.56567
from bnraph going low
FR,CHI,CHB,CHL 0.120885292
6689.95567
from bnraph going low
FR,CHI,CHB,CHL 0.100000000E-01 6677.24298
from bnraph going low
FR,CHI,CHB,CHL 0.100000000E-01 6639.01696
from bnraph going low
FR,CHI,CHB,CHL 0.100000000E-01 6694.62463
IREV=
1 IEND=99993
from bnraph going low
FR,CHI,CHB,CHL 0.901000000
6694.62463
IREV=
2 IEND=99993
from bnraph going low
FR,CHI,CHB,CHL 0.990100000
6694.62463
0.100000000E+67 0.500000000E+67
7007.78950
9612.87997
6599.35985
7079.62515
6691.10203
6713.67300
6705.41629
6711.56567
6685.54279
6689.95567
6674.19657
6677.24298
6635.20438
6639.01696
6635.20438
6639.01696
6635.20438
6639.01696
IREV=
20 IEND=99993
FR,CHI,CHB,CHL 0.999010000
6639.01696
6639.01696
6639.01696
AT END FR, CHI 0.999010000
6639.01696
TEMP2= 0.000000000E+00
NEW SECTION FR=
0.9990100000000001
FR,CHI,CHB,CHL 0.999999990
6639.01696
0.100000000E+67 0.500000000E+67
FR,CHI,CHB,CHL 0.999999970
6639.01696
6639.01676
6639.01696  At this
IREV=
1 IEND=99992
level there are no small predictions – rather than
FR,CHI,CHB,CHL 0.999999997
6639.01696
6639.01694
6639.01696 the
IREV=
2 IEND=99992
traditional fr  1, would do better with fr  .9 so
FR,CHI,CHB,CHL 0.999999999
6639.01696
6639.01695
6639.01696 that a larger
AT END FR, CHI 0.999999999
6639.01696
distance in parameter space is
TEMP2= 0.999999953E-09
attempted.
NEW SECTION FR=
0.9999999990000000
Figure 3 (fi-fA(xi))/(fi  1e-15)
The systematic errors appear to be on the order of 5  10-16. The “random” errors
result in function errors that are reduced from the fluctuationa above by a factor
approximately equal to 5000/18 = 16 so that these are also less than 1  10-15.
 Following is last constant file with all updates 
aigau.dir, OUTPUT FILE NAME
..\TestData\brack.OUT,
DATA SET NAME
1,
DIMENSION OF X IN F(X)
9,9 NUMBER OF NUMERATOR TERMS, DENOMINATOR TERMS
0, 0 fits to exp(-x**2) times a Pade, 0 to a Pade
1E-30,1E-15, err(i)= max(ep1,ep2*dat(i))
100000,
NUMBER OF MINIMIZATION STEPS
.9, INITIAL FR DECREASE DESIRED
1 VARY O FIX
FITTED CONS ERROR IN CONS
0 0.5000000000000000
+- 0.000E+00
1 -1.065781843119121
1
1.098849939447941
1 -0.7059835832256011
1 0.3064328550407249
1 -0.9175866513292681E-01
1 0.1853701251033871E-01
1 -0.2323566331513835E-02
1 0.1400748062111785E-03
1 -3.259942853333778
1
4.876151480518089
1 -4.406424837428250
1
2.661133284964304
1 -1.118888036108821
1 0.3293940293195529
1 -0.6596029956005756E-01
1 0.8236827162542478E-02
1 -0.4965522790119376E-03
CHIS= 6368.
FOR 5001 DATA
CHI USED IN CALCULATING ERRORS
+- 0.254E-01
+- 0.441E-01
+- 0.380E-01
+- 0.202E-01
+- 0.709E-02
+- 0.163E-02
+- 0.228E-03
+- 0.152E-04
+- 0.507E-01
+- 0.145
+- 0.189
+- 0.147
+- 0.746E-01
+- 0.256E-01
+- 0.581E-02
+- 0.810E-03
+- 0.540E-04
POINTS
IS 6368.
These constants above give
7
AiGauss( z )  exp   z 2 
c z
I 0
8
I 1
I
1   cI  7 z I
 1015
I 1
Note that the signs of the constants are such that for negative z, every term adds. There is
no cancellation and the truncation error is minimized. There is a slight gain in fitting the
results of the integral directly. The constants below are slightly better.
Exp(-z2)Brack(z)
The code was designed to fit AiGauss directly. Starting with the constants above,
a number of formats needed changing in ..\TestData\genint\TGLEG67.FOR to get the full
range of the data – The data above with much smaller derivatives seems unaffected by
these changes. The final fit begins with the constants from fitting brack above and a chisquare of 4647. This is lower than that for brack. The values used now are direct
integration values while brack’s fit multiplied these values by exp(z2). It appears that this
introduces an extra 2000 into the chi-square.
aigau2.dir, OUTPUT FILE NAME
..\TestData\aigauss.OUT,
DATA SET NAME
1,
DIMENSION OF X IN F(X)
9,9 NUMBER OF NUMERATOR TERMS, DENOMINATOR TERMS
1, 1 fits to exp(-x**2) times a Pade, 0 to a Pade
1E-30,1E-15, err(i)= max(ep1,ep2*dat(i))
100000,
NUMBER OF MINIMIZATION STEPS
.9, INITIAL FR DECREASE DESIRED
1 VARY O FIX
FITTED CONS ERROR IN CONS
0 0.5000000000000000
+- 0.000E+00
1 -1.502885941371286
+- 0.652E-01
1
1.851590712127221
+- 0.113
1 -1.350498684163228
+- 0.964E-01
1 0.6470300251477153
+- 0.510E-01
1 -0.2101536907945629
+- 0.178E-01
1 0.4549469611716167E-01
1 -0.6057434494797435E-02
1 0.3854291140973252E-03
1 -4.134151049838160
1
7.368071342513580
1 -7.633077301200458
1
5.148920726637210
1 -2.373642689043555
1 0.7557095175197603
1 -0.1619578118794217
1 0.2147303973187547E-01
1 -0.1366310760644399E-02
CHIS= 4580.
FOR 5001 DATA
CHI USED IN CALCULATING ERRORS
+- 0.405E-02
+- 0.562E-03
+- 0.371E-04
+- 0.130
+- 0.372
+- 0.482
+- 0.372
+- 0.188
+- 0.640E-01
+- 0.144E-01
+- 0.199E-02
+- 0.131E-03
POINTS
IS 4984.
The minimum without the truncation correction was 4519. Simply re-reading the
constants goes to 4619  4566 (This is the random/truncation zone). The constants
above are the best for representing AiGauss.
Figure 4 Final fit to AiGauss. -- Black is data, blue is fit, red is |data-fit|
Figure 5 (data-fit)/(data 10-15)
There is a truncation like error possibly in the Gauss Quadrature for -5.5<z<-3.5.
There is a small systematic effect at about -2.0 of a size < 1. In general the “random”
error in the fit is less than that in the data by a factor of (5000/18) = 16.7  that the
error is < 10-15.
The values below -6 are quickly generated to this accuracy using the asymptotic
expansion ..\..\TestData\genint\Expanding the integrand.htm. doc. -..\..\TestData\genint\genint.zip Those above 0 are 1-the value below zero. Thus the code
to find AiGauss everywhere is ready to write.
Using the files
The code for making this fit is in ..\..\zips\FittingBrack.zip
Directory of FittingBrack.zip
aigau2.dir
-- sample direction file for fitting brack.out
AIGAUSS.OUT -- data of the form exp(-z2)*Brack(z) generated by direct integration of
Gaussian
BRACK.OUT -- data of the form Brack(z) Integral has been multiplied by exp(z2) with
slight loss of accuracy
transmp.exe -- Bailey’s code for converting to multi-precision -..\..\..\..\MultiplePrecision\bailey\Welcome.htm
mpfun.for --- Bailey’s set of multi-precision routines
nlfit.wpj Watcom ide’s for making nlfit
nlfit.tgt
cnlfit.bat -- bat file for using wfl386 to make nlfit
mpnlfit.wpj Watcom ide’s for making mpnlfit
mpnlfit.tgt
cmpnlfit.bat – bat file for using wfl386 to make mpnlfit
Fortran files
MAOPEN.FOR -- used by Watfor
MSOPEN.FOR – used by watcom
WSYSTEM.FOR -- used by Watfor
openwsys.for – used by watcom
nlfpade.for - nlfit slightly modified to have Nnum, Nden, eps1, and eps2. Passes flag
for fitting Brack or exp*Brack to poly
MPnlfpade.for – Generated from nlfpade by running transmp.exe
ROBMIN.FOR --- minimization code. Run transmp.exe and enter robmin.for to
produce
MProbmin.for --- MProbmin
polyexpa.for --- The poly routine for making exp(x2)Brack(x) or simply Brack(x)